Singular Matrix

A singular matrix is a square matrix of determinant “0.” i.e., a square matrix A is singular if and only if det A = 0. Inverse of a matrix A is found using the formula A^-1 = (adj A) / (det A). Thus, a matrix is called a square matrix if its determinant is zero. Now let us discuss about singular matrix, its properties, and others in detail.

What is a Singular Matrix?

A square matrix is said to be a singular matrix if its determinant is zero and it is not invertible. In a singular matrix, some rows and columns are linearly dependent. As the rows and columns of a singular matrix are linearly dependent, the rank of the matrix will be less than the order of the matrix.

The image given below is an “m × n” matrix that has “m” rows and “n” columns.

We know that the formula to determine the inverse of a matrix is equal to the adjoint of the matrix divided by the determinant of the matrix, i.e., A^-1 = (adj A) / |A|. From the definition of a singular matrix, we know that |A| = 0, so its inverse is not defined.

Let us consider that A and B are two square matrices of order “n × n”

If,

AB = BA = I

where,

• I is an identity or unit matrix of order n
• B is said to be the inverse matrix of A

Thus, matrix A is a non-singular matrix.

Properties of a Singular Matrix

The following are the properties of the Singular Matrix:

• Every singular matrix must be a square matrix, i.e., a matrix that has an equal number of rows and columns.
• The determinant of a singular matrix is equal to zero.
• As the determinant of a singular matrix is zero, its inverse is not defined.
• A zero matrix of any order matrix is a singular matrix, as its determinant is zero.
• In a singular matrix, some rows and columns are linearly dependent.
• The rank of a singular matrix will be less than the order of the matrix, i.e., Rank (A) < Order of A.
• A matrix that has any two rows or any two columns identical is singular, as the determinant of such a matrix is zero.
• When a row or column’s elements in a matrix are all zeros, then the matrix is singular, as its determinant is zero.
• When one row (or column) of a matrix is a scalar multiple of another row (or column), then the matrix is singular as its determinant is zero. 

Differences Between Singular and Non-Singular Matrix

Differences between Singular Matrix and Non-Singular Matrix can be understood using the table given below

Singular Matrix Vs Non-Singular Matrix
Singular Matrix	Non-Singular Matrix
A square matrix is said to be a singular matrix if its determinant is zero, i.e., det A = 0.	A square matrix is said to be a non-singular matrix if its determinant is not zero, i.e., det A ≠ 0.
If a matrix is singular, then its inverse is not defined.	If a matrix is non-singular, then its inverse is defined.
The rank of a singular matrix will be less than the order of the matrix, i.e., Rank (A) < Order of A.	The rank of a non-singular matrix will be equal to the order of the matrix, i.e., Rank (A) = Order of A.
In a singular matrix, some rows and columns are linearly dependent.	In a non-singular matrix, all the rows and columns are linearly independent.
[Tex]A = \left(\begin{array}{ccc} 2 & 2 & 4\\ 1 & 1 & 2\\ 3 & 7 & 9 \end{array}\right) [/Tex]	[Tex]B = \left[\begin{array}{ccc} 1 & 2 & -3\\ 6 & 0 & 8\\ -1 & 4 & 0 \end{array}\right] [/Tex]

Identifying a Singular Matrix

Follow the conditions given below to determine whether the given matrix is singular or not.

• Determine whether the given matrix is a square matrix or not.
• If the given matrix is a square matrix, then find the determinant of the matrix.

⇒ If |A|= 0, then the given matrix is singular.

⇒ If |A|≠0, then the given matrix is non-singular.

Formula for Determinant of “2 × 2” Matrix

If A = [Tex]\left[\begin{array}{cc} a & b\\ c & d \end{array}\right]     [/Tex] is a “2 × 2” matrix, then its determinant is 

|A|= [ad – bc]

Formula for Determinant of “3 × 3” Matrix

If A = [Tex]\left[\begin{array}{ccc} a_{1} & a_{2} & a_{3}\\ b_{1} & b_{2} & b_{3}\\ c_{1} & c_{2} & c_{3} \end{array}\right]     [/Tex] is a “3 × 3” matrix, then its determinant is 

|A|= a₁(b₂c₃ – b₃c₂) – a₂(b₁c₃ – b₃c₁) + a₃(b₁c₂ – b₂c₁)

Solved Examples on Singular Matrix

Example 1: Find the value of k if the matrix given below, is a singular matrix.

[Tex]A = \left[\begin{array}{cc} k & -4\\ 5 & 2 \end{array}\right] [/Tex]

Solution:

Given matrix A = [Tex]\left[\begin{array}{cc} k & -4\\ 5 & 2 \end{array}\right] [/Tex]

We know that the determinant of a singular matrix is zero, i.e., det A = 0

⇒ (2×k) – (–4 × 5) = 0

⇒ 2k + 20 = 0

⇒ 2k = -20

⇒ k = –20/2 = –10

Hence, the value of k if the given matrix is a singular matrix is –10.

Example 2: Determine the inverse of the matrix given below.

[Tex]P = \left[\begin{array}{cc} -3 & 4\\ 6 & -8 \end{array}\right] [/Tex]

Solution:

Given matrix [Tex] P = \left[\begin{array}{cc} -3 & 4\\ 6 & -8 \end{array}\right] [/Tex]

P^-1 = Adj P / |P|

Now, let us find the determinant of the matrix P.

|P| = (–3 × –8) – (6 × 4)

|P| = 24 – 24 = 0

Since, the determinant of matrix P = 0, it is a singular matrix, and its inverse matrix doesn’t exist.

Example 3: Determine whether the given matrix is singular or not.

[Tex]A = \left[\begin{array}{ccc} 1 & 0 & -3\\ 0 & 5 & 2\\ -1 & 4 & 0 \end{array}\right] [/Tex]

Solution:

Given matrix A = [Tex]\left[\begin{array}{ccc} 1 & 0 & -3\\ 0 & 5 & 2\\ -1 & 4 & 0 \end{array}\right] [/Tex]

To determine whether the given matrix is singular or not, we have to find its determinant.

det A = 1[(5 × 0) – (4 × 2)] – 0[(0 × 0) – (2 × –1)] + (-3) [(0 × 4) – (–1 × 5)]

⇒ |A| = (1 × -8) – 0 + (–3 × 5) 

⇒ |A| = –8 – 15 = –23 ≠ 0

Since the determinant of the given matrix is not equal to zero, it is a non-singular matrix.

Example 4: Find the value of b if the matrix given below, is a singular matrix.

[Tex]B = \left[\begin{array}{cc} 9 & b\\ 6 & -2 \end{array}\right] [/Tex]

Solution:

Given matrix [Tex]B = \left[\begin{array}{cc} 9 & b\\ 6 & -2 \end{array}\right] [/Tex]

We know that the determinant of a singular matrix is zero, i.e., det B = 0

⇒ (9 × –2) – (6 × b) = 0

⇒ –18 – 6b = 0

⇒ –6b = 18

⇒ b = 18/–6 = –3

Hence, the value of b if the given matrix is a singular matrix is –3.

FAQs on Singular Matrix

1. Define a Matrix.

A matrix is defined as a rectangular array of numbers that are arranged in rows and columns.

2. What is a Singular Matrix?

A square matrix is said to be a singular matrix if its determinant is zero and it is not invertible.

3. What is the Rank of a Singular Matrix of Order “3 × 3”?

If the given matrix A is singular, then its determinant is zero. Now, the rank of the given matrix will be less than the order of the matrix, i.e., rank (A) < 3.

4. What is the Determinant of a Singular Matrix?

The determinant of a matrix determines whether it is singular or non-singular. So, a matrix is said to be singular if its determinant is zero.

5. Is a Zero Matrix a Singular Matrix?

As the determinant of a singular matrix is zero, it is a singular matrix.

6. What is Rank of a Singular Matrix?

The rank of singular matrix ‘n’ is always less than ‘n’.